Holomorphic automorphisms of Markov-type surfaces

TL;DR

This paper characterizes the holomorphic automorphisms of Markov-type surfaces, establishing a symplectic Andersén–Lempert theorem and demonstrating how these automorphisms can be realized via injective maps of Markov triples.

Contribution

It introduces a singular symplectic version of the Andersén–Lempert theorem for complex surfaces and describes their automorphisms, connecting algebraic and symplectic geometry.

Findings

Proves the symplectic density property for Markov-type surfaces

Describes holomorphic symplectic automorphisms explicitly

Shows injective maps of Markov triples correspond to automorphisms

Abstract

Every complex surface of Markov type, i.e.\ the variety given by $x^2 + y^2 + z^2 + Exyz - Ax - By - Cz - D = 0$, has the symplectic density property and the Hamiltonian density property. We prove a singular symplectic version of the Anders{\'e}n--Lempert theorem for normal reduced affine complex varieties and apply it to describe the holomorphic symplectic automorphisms of a complex surface of Markov type. To this end, we also investigate the germs of vector fields in isolated singularities of type $A_k$ and $D_k$. Moreover, we show that any injective self-map of the set of ordered Markov triples can be realized by a holomorphic symplectic automorphism.